009B Sample Final 1 - Revision history

Kayla Murray: /* Problem 1 */

2017-02-27T20:19:35Z

Problem 1

2017-02-27T20:12:58Z

Problem 5

2017-02-27T20:08:00Z

Problem 3

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Problem 4

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Problem 7

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Problem 6

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Problem 5

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Problem 4

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Problem 3

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Problem 2

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 == [[009B_Sample Final 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam">Consider the region bounded by the following two functions:
+<span class="exam">Suppose the speed of a bee is given in the table.
-<span class="exam">(a) Using the lower sum with three rectangles having equal width, approximate the area.
+<table border="1" cellspacing="0" cellpadding="6" align = "center">
-<span class="exam">(b) Using the upper sum with three rectangles having equal width, approximate the area.
+</table>
-<span class="exam">(c) Find the actual area of the region.
+<span class="exam">(a) Using the given measurements, find the left-hand estimate for the distance the bee moved during this experiment.
 == [[009B_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==

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 == [[009B_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> Consider the solid obtained by rotating the area bounded by the following three functions about the <math style="vertical-align: -3px">y</math>-axis:
+<span class="exam"> The region bounded by the parabola &nbsp;<math style="vertical-align: -4px">y=x^2</math>&nbsp; and the line &nbsp;<math style="vertical-align: -4px">y=2x</math>&nbsp; in the first quadrant is revolved about the &nbsp;<math style="vertical-align: -4px">y</math>-axis to generate a solid.
-::<span class="exam"> <math style="vertical-align: -4px">x=0,y=e^x,</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=ex</math>.
+<span class="exam">(a) Sketch the region bounded by the given functions and find their points of intersection.
 <span class="exam">(b) Set up the integral for the volume of the solid.

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 == [[009B_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam">Consider the area bounded by the following two functions:
-::<span class="exam"><math style="vertical-align: -4px">y=\sin x</math>&nbsp; and &nbsp;<math style="vertical-align: -13px">y=\frac{2}{\pi}x</math>.
+::<span class="exam"><math style="vertical-align: -4px">y=\cos x</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=2-\cos x,~0\le x\le 2\pi.</math>
-<span class="exam">(a) Find the three intersection points of the two given functions. (Drawing may be helpful.)
+<span class="exam">(a) Sketch the graphs and find their points of intersection.
 <span class="exam">(b) Find the area bounded by the two functions.

@@ Line 37: / Line 37: @@
 <span class="exam"> Compute the following integrals.
-<span class="exam">(a) &nbsp;<math>\int e^x(x+\sin(e^x))~dx</math>
+<span class="exam">(a) &nbsp;<math>\int \frac{t^2}{\sqrt{1-t^6}}~dt</math>
 <span class="exam">(b) &nbsp;<math>\int \frac{2x^2+1}{2x^2+x}~dx</math>

← Older revision		Revision as of 17:15, 27 February 2017
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	::<math>y=1-x^2,~~~0\leq x \leq 1</math>		::<math>y=1-x^2,~~~0\leq x \leq 1</math>

−	<span class="exam">is rotated about the <math style="vertical-align: -3px">y</math>-axis. Find the area of the resulting surface.	+	<span class="exam">is rotated about the  <math style="vertical-align: -3px">y</math>-axis. Find the area of the resulting surface.

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 <span class="exam"> Evaluate the improper integrals:
-<span class="exam">(a) <math>\int_0^{\infty} xe^{-x}~dx</math>
+<span class="exam">(a) &nbsp;<math>\int_0^{\infty} xe^{-x}~dx</math>
-<span class="exam">(b) <math>\int_1^4 \frac{dx}{\sqrt{4-x}}</math>
+<span class="exam">(b) &nbsp;<math>\int_1^4 \frac{dx}{\sqrt{4-x}}</math>
 == [[009B_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==

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 <span class="exam"> Consider the solid obtained by rotating the area bounded by the following three functions about the <math style="vertical-align: -3px">y</math>-axis:
-::<span class="exam"> <math style="vertical-align: 0px">x=0</math>, <math style="vertical-align: -4px">y=e^x</math>, and <math style="vertical-align: -4px">y=ex</math>.
+::<span class="exam"> <math style="vertical-align: -4px">x=0,y=e^x,</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=ex</math>.
 <span class="exam">(a) Sketch the region bounded by the given three functions. Find the intersection point of the two functions:
-::<span class="exam"><math style="vertical-align: -4px">y=e^x</math> and <math style="vertical-align: -4px">y=ex</math>. (There is only one.)
+::<span class="exam"><math style="vertical-align: -4px">y=e^x</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=ex</math>. (There is only one.)
 <span class="exam">(b) Set up the integral for the volume of the solid.

@@ Line 18: / Line 18: @@
 ::<math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t\,dt\bigg).</math>
-<span class="exam">(a) Compute <math style="vertical-align: -15px">f(x)=\int_{-1}^{x} \sin(t^2)2t\,dt</math>.
+<span class="exam">(a) Compute &nbsp;<math style="vertical-align: -15px">f(x)=\int_{-1}^{x} \sin(t^2)2t\,dt</math>.
-<span class="exam">(b) Find <math style="vertical-align: -5px">f'(x)</math>.
+<span class="exam">(b) Find &nbsp;<math style="vertical-align: -5px">f'(x)</math>.
 <span class="exam">(c) State the Fundamental Theorem of Calculus.
-<span class="exam">(d) Use the Fundamental Theorem of Calculus to compute&thinsp; <math style="vertical-align: -15px">\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t\,dt\bigg)</math> &thinsp;without first computing the integral.
+<span class="exam">(d) Use the Fundamental Theorem of Calculus to compute &nbsp;<math style="vertical-align: -15px">\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t\,dt\bigg)</math>&nbsp; without first computing the integral.
 == [[009B_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==